1. Find the 4-tuples (a, b, c, d) that solves the system of linear equations

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1 MÄLARDALEN UNIVERSITY School of Education, Culture and Communication Department of Applied Mathematics Examiner: Lars-Göran Larsson EXAMINATION IN MATHEMATICS MAA150 Vector Algebra, TEN1 Date: Write time: 3 hours Aid: Writing materials, ruler This examination is intended for the examination part TEN1. The examination consists of five randomly ordered problems each of which is worth at maximum 5 points. The pass-marks 3, 4 and 5 require a minimum of 12, 16 and 21 points respectively. The minimum points for the ECTS-marks E, D, C, B and A are 12, 13, 16, 20 and 24 respectively. If the obtained sum of points is denoted S 1, and that obtained at examination TEN1 S 2, the marks for a completed course are determined according to S 1, S 2 12 and S 1 + 2S S 1, S 2 12 and S 1 + 2S 2 38 E S 1, S 2 12 and 48 S 1 + 2S S 1, S 2 12 and 39 S 1 + 2S 2 47 D 63 S 1 + 2S 2 5 S 1, S 2 12 and 48 S 1 + 2S 2 59 C S 1, S 2 12 and 60 S 1 + 2S 2 71 B 72 S 1 + 2S 2 A Solutions are supposed to include rigorous justifications and clear answers. All sheets of solutions must be sorted in the order the problems are given in. 1. Find the 4-tuples (a, b, c, d) that solves the system of linear equations 5a 2b 7c + 8d = 0, 2a + 3b + c + 7d = 0, 3a 5b 8c + d = 0, 4a + b 3c + 9d = Find an equation on parameter-free form for the plane π which contains the point P 1 : (3, 4, 2), is parallel with the line λ 2 : (x, y, z) = (3 + 5t, 2t, 4 3t), and is orthogonal to the plane π 3 : x + 3y + z + 2 = 0. It is assumed that the standard basis is a right-handed ON-basis. 3. Find the matrix A that solves the equation ( ) T ( ) = A + A Find the coordinates of the point Q which belongs to the line λ : (x, y, z) = (1 2t, 2 t, 3 + t) and which is closest to the point P : (2, 1, 5). It is assumed that the standard basis is an ON-basis. 5. Find, in polar form, the complex number w which solves the equation i + 3 i + 3 = 1 w (3i 1), where w denotes the complex conjugate of w. Om du föredrar uppgifterna formulerade på svenska, var god vänd på bladet.

2 MÄLARDALENS HÖGSKOLA Akademin för utbildning, kultur och kommunikation Avdelningen för tillämpad matematik Examinator: Lars-Göran Larsson TENTAMEN I MATEMATIK MAA150 Vektoralgebra, TEN1 Datum: Skrivtid: 3 timmar Hjälpmedel: Skrivdon, linjal Denna tentamen är avsedd för examinationsmomentet TEN1. Provet består av fem stycken om varannat slumpmässigt ordnade uppgifter som vardera kan ge maximalt 5 poäng. För godkänd-betygen 3, 4 och 5 krävs erhållna poängsummor om minst 12, 16 respektive 21 poäng. Om den erhållna poängen benämns S 1, och den vid tentamen TEN2 erhållna S 2, bestäms graden av ett sammanfattningsbetyg på en slutförd kurs enligt S 1, S 2 12 och S 1 + 2S S 1, S 2 12 och 48 S 1 + 2S S 1 + 2S 2 5 Lösningar förutsätts innefatta ordentliga motiveringar och tydliga svar. Samtliga lösningsblad skall vid inlämning vara sorterade i den ordning som uppgifterna är givna i. 1. Bestäm de 4-tipplar (a, b, c, d) som löser det linjära ekvationssystemet 5a 2b 7c + 8d = 0, 2a + 3b + c + 7d = 0, 3a 5b 8c + d = 0, 4a + b 3c + 9d = Bestäm en ekvation på parameterfri form för det plan π som innehåller punkten P 1 : (3, 4, 2), är parallellt med linjen λ 2 : (x, y, z) = (3 + 5t, 2t, 4 3t), och är ortogonalt mot planet π 3 : x + 3y + z + 2 = 0. Det antages att standardbasen är en högerorienterad ON-bas. 3. Bestäm den matris A som löser ekvationen ( ) T ( ) = A + A Bestäm koordinaterna för den punkt Q som tillhör linjen λ : (x, y, z) = (1 2t, 2 t, 3 + t) och som ligger närmast punkten P : (2, 1, 5). Det antages att standardbasen är en ON-bas. 5. Bestäm, på polär form, det komplexa tal w som löser ekvationen i + 3 i + 3 = 1 w (3i 1), där w betecknar komplexkonjugatet till w. If you prefer the problems formulated in English, please turn the page.

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5 MÄLARDALEN UNIVERSITY School of Education, Culture and Communication Department of Applied Mathematics Examiner: Lars-Göran Larsson Final examination TEN (aa, bb, cc, dd) = (rr 2ss, rr ss, rr, ss) where rr, ss RR EXAMINATION IN MATHEMATICS MAA150 Vector algebra EVALUATION PRINCIPLES with POINT RANGES Academic year: 2017/18 Maximum points for subparts of the problems in the final examination 2p: Correctly found at least the echelon form of the coefficient matrix in preparation for finding the 4-tuples 1p: Correctly, from the (row-reduced) echelon form, identified the correponding system of linear equations 2p: Correctly found the general 4-tuple solving the system of linear equations 2. ππ 11xx 8yy + 13zz 91 = 0 1p: Correctly concluded that a normal of the plane ππ is a vector nn which is orthogonal to the line λλ 2 and to a normal of the plane ππ 3 2p: Correctly identified a vector vv 2 parallel with λλ 2, and a vector nn 3 normal to ππ 3, and correctly worked out a vector nn orthogonal to the vectors vv 2 and nn 3 2p: Correctly formulated an equation for the plane ππ on parameter-free form One scenario p: Correctly factorized the RHS of the equation 1p: Correctly transposed the matrix of the LHS of the equation, and from the resulting equation C = AB correctly found that A = CB 1 1p: Correctly found (the entries of) the matrix B 1 1p: Correctly found (the entries of) the matrix A Another scenario p: Correctly proposed the entries of the matrix A and correctly worked out the matrix operations of the LHS and the RHS of the equation 2p: Correctly solved the system of (four) linear equations, and correctly put together the matrix A 4. QQ 0, 3, 7 2p: Correctly concluded that the point QQ can be addressed by 2 2 adding to the coordinates of a point PP 0 of the line, the coordinates of the ortogonal projection of a vector uu, represented by the directed line segment from the point PP 0 to the point PP, on a vector vv parallell with the line 1p: Correctly identified vectors uu and vv 1p: Correctly found the orthogonal projection of uu on vv 1p: Correctly found the coordinates of the point QQ 5. ww = 1 2 cos 2ππ 3 + ii sin 2ππ 3 = 1 2 eeii2ππ 3 3p: Correctly found ww on the form aa + bbbb, aa, bb RR, where a correct treatment of the involved complex conjugation is worth one (1) of the three points 1p: Correctly found a representative argument of ww 1p: Correctly found the absolute value of ww, and correctly written ww in polar form 1 (1)